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Vol 83, No 5 (2019)
Articles
Conformally invariant inequalities in domains in Euclidean space
Abstract
We study conformally invariant integral inequalities for real-valuedfunctions defined on domains $\Omega$ in $n$-dimensional Euclideanspace. The domains considered are of hyperbolic type, that is, they admita hyperbolic radius $R=R(x, \Omega)$ satisfying the Liouvillenon-linear differential equation and vanishing on the boundary of thedomain. We prove several inequalities which hold for all smooth compactlysupported functions $u$ defined on a given domain of hyperbolic type.Here are two of them:$$\int|\nabla u|^2R^{2-n} dx \geq n (n-2)\int|u|^2R^{-n} dx,$$$$\int|(\nabla u, \nabla R)|^p R^{p-s} dx\geq \frac{2^pn^p}{p^p}\int|u|^pR^{-s} dx,$$where $n\geq 2$, $1\leq p< \infty$ and $1+n/2 \leq s <\infty$.We also study the relations between Euclidean and hyperbolic characteristicsof domains.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(5):3-26
3-26
Adaptive energy-saving approximation for stationary processes
Abstract
We consider a stationary process (with either discrete or continuous time)and find an adaptive approximating stationary process combining highquality approximation and other good propertiesthat can be interpretedas additional smoothness or small expense of energy. The problem is solvedin terms of spectral characteristics of the original process usingthe classical analytic methods of prediction theory.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(5):27-52
27-52
53-87
Almost solubility of classes of non-linear integral equations of the first kind on cones
Abstract
Using convexity properties of the images of completely continuousnon-linear integral operators, we describe the closed convex cones lyingeither in the recessive cone, or in the tangent cone of the closed imageof the operator being studied (depending on the nature of the integrand).These cones are determined by the principal part of theasymptotics of the integrand at infinity, independently of the variation ofthe subordinate part. We discuss applications to the generalized solubilityof non-linear integral equations of the first kind.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(5):88-106
88-106
107-148
Properties of factorization operators in boundary crossing problems for random walks
Abstract
We study the properties of operators arising in the calculation ofdouble Laplace–Stieltjes transforms of distributions in variousboundary crossing problems for random walks. Such operators aredefined in terms of the components of the Wiener–Hopf factorization.We give bounds for the norms of these operators and prove continuitytheorems.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2019;83(5):149-166
149-166
167-180